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A (smooth) Riemannian manifold is a smooth manifold without boundary, equipped with a Riemannian metric , which assigns a length to every tangent vector at a point , and more generally assigns an inner product
to every pair of tangent vectors at a point . (We use Roman font for here, as we will need to use to denote group elements later in this post.) This inner product is assumed to symmetric, positive definite, and smoothly varying in , and the length is then given in terms of the inner product by the formula
In coordinates (and also using abstract index notation), the metric can be viewed as an invertible symmetric rank tensor , with
One can also view the Riemannian metric as providing a (self-adjoint) identification between the tangent bundle of the manifold and the cotangent bundle ; indeed, every tangent vector is then identified with the cotangent vector , defined by the formula
In coordinates, .
A fundamental dynamical system on the tangent bundle (or equivalently, the cotangent bundle, using the above identification) of a Riemannian manifold is that of geodesic flow. Recall that geodesics are smooth curves that minimise the length
There is some degeneracy in this definition, because one can reparameterise the curve without affecting the length. In order to fix this degeneracy (and also because the square of the speed is a more tractable quantity analytically than the speed itself), it is better if one replaces the length with the energy
Minimising the energy of a parameterised curve turns out to be the same as minimising the length, together with an additional requirement that the speed stay constant in time. Minimisers (and more generally, critical points) of the energy functional (holding the endpoints fixed) are known as geodesic flows. From a physical perspective, geodesic flow governs the motion of a particle that is subject to no external forces and thus moves freely, save for the constraint that it must always lie on the manifold .
One can also view geodesic flows as a dynamical system on the tangent bundle (with the state at any time given by the position and the velocity ) or on the cotangent bundle (with the state then given by the position and the momentum ). With the latter perspective (sometimes referred to as cogeodesic flow), geodesic flow becomes a Hamiltonian flow, with Hamiltonian given as
where is the inverse inner product to , which can be defined for instance by the formula
In coordinates, geodesic flow is given by Hamilton’s equations of motion
In terms of the velocity , we can rewrite these equations as the geodesic equation
If the manifold is an embedded submanifold of a larger Euclidean space , with the metric on being induced from the standard metric on , then the geodesic flow equation can be rewritten in the equivalent form
where is now viewed as taking values in , and is similarly viewed as a subspace of . This is intuitively obvious from the geometric interpretation of geodesics: if the curvature of a curve contains components that are transverse to the manifold rather than normal to it, then it is geometrically clear that one should be able to shorten the curve by shifting it along the indicated transverse direction. It is an instructive exercise to rigorously formulate the above intuitive argument. This fact also conforms well with one’s physical intuition of geodesic flow as the motion of a free particle constrained to be in ; the normal quantity then corresponds to the centripetal force necessary to keep the particle lying in (otherwise it would fly off along a tangent line to , as per Newton’s first law). The precise value of the normal vector can be computed via the second fundamental form as , but we will not need this formula here.
In a beautiful paper from 1966, Vladimir Arnold (who, sadly, passed away last week), observed that many basic equations in physics, including the Euler equations of motion of a rigid body, and also (by which is a priori a remarkable coincidence) the Euler equations of fluid dynamics of an inviscid incompressible fluid, can be viewed (formally, at least) as geodesic flows on a (finite or infinite dimensional) Riemannian manifold. And not just any Riemannian manifold: the manifold is a Lie group (or, to be truly pedantic, a torsor of that group), equipped with a right-invariant (or left-invariant, depending on one’s conventions) metric. In the context of rigid bodies, the Lie group is the group of rigid motions; in the context of incompressible fluids, it is the group ) of measure-preserving diffeomorphisms. The right-invariance makes the Hamiltonian mechanics of geodesic flow in this context (where it is sometimes known as the Euler-Arnold equation or the Euler-Poisson equation) quite special; it becomes (formally, at least) completely integrable, and also indicates (in principle, at least) a way to reformulate these equations in a Lax pair formulation. And indeed, many further completely integrable equations, such as the Korteweg-de Vries equation, have since been reinterpreted as Euler-Arnold flows.
From a physical perspective, this all fits well with the interpretation of geodesic flow as the free motion of a system subject only to a physical constraint, such as rigidity or incompressibility. (I do not know, though, of a similarly intuitive explanation as to why the Korteweg de Vries equation is a geodesic flow.)
One consequence of being a completely integrable system is that one has a large number of conserved quantities. In the case of the Euler equations of motion of a rigid body, the conserved quantities are the linear and angular momentum (as observed in an external reference frame, rather than the frame of the object). In the case of the two-dimensional Euler equations, the conserved quantities are the pointwise values of the vorticity (as viewed in Lagrangian coordinates, rather than Eulerian coordinates). In higher dimensions, the conserved quantity is now the (Hodge star of) the vorticity, again viewed in Lagrangian coordinates. The vorticity itself then evolves by the vorticity equation, and is subject to vortex stretching as the diffeomorphism between the initial and final state becomes increasingly sheared.
The elegant Euler-Arnold formalism is reasonably well-known in some circles (particularly in Lagrangian and symplectic dynamics, where it can be viewed as a special case of the Euler-Poincaré formalism or Lie-Poisson formalism respectively), but not in others; I for instance was only vaguely aware of it until recently, and I think that even in fluid mechanics this perspective to the subject is not always emphasised. Given the circumstances, I thought it would therefore be appropriate to present Arnold’s original 1966 paper here. (For a more modern treatment of these topics, see the books of Arnold-Khesin and Marsden-Ratiu.)
In order to avoid technical issues, I will work formally, ignoring questions of regularity or integrability, and pretending that infinite-dimensional manifolds behave in exactly the same way as their finite-dimensional counterparts. In the finite-dimensional setting, it is not difficult to make all of the formal discussion below rigorous; but the situation in infinite dimensions is substantially more delicate. (Indeed, it is a notorious open problem whether the Euler equations for incompressible fluids even forms a global continuous flow in a reasonable topology in the first place!) However, I do not want to discuss these analytic issues here; see this paper of Ebin and Marsden for a treatment of these topics.